A363600 Number of partitions of n into distinct parts where there are k^2+1 kinds of part k.
1, 2, 6, 20, 52, 140, 356, 880, 2123, 5016, 11610, 26400, 59130, 130476, 284216, 611592, 1301344, 2740194, 5713930, 11806144, 24184908, 49142504, 99091244, 198360536, 394342884, 778818658, 1528531702, 2982017956, 5784365082, 11158728448, 21413292868
Offset: 0
Programs
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PARI
my(N=40, x='x+O('x^N)); Vec(prod(k=1, N, (1+x^k)^(k^2+1)))
Formula
G.f.: Product_{k>=1} (1+x^k)^(k^2+1).
a(0) = 1; a(n) = (-1/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d) * d * (d^2+1) ) * a(n-k).
Extensions
Name suggested by Joerg Arndt, Jun 11 2023